Method and apparatus for distributing a quantum key

ABSTRACT

A method for distributing a quantum key is provided, including sending a first photon to a first receiver; sending a second photon to a second receiver, the first and second photons being a pair of time-energy entangled photons; and providing a coding scheme comprising a plurality of time bins and a plurality of frequency bins, wherein a combination of a time bin and a frequency bin corresponds to a character.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.13/646,545 filed Oct. 5, 2012, which claims priority to U.S. provisionalapplication 61/543,691 filed Oct. 5, 2011, which is incorporated byreference in its entirety herein.

FIELD

The disclosed subject matter generally relates to integrated opticalsystems. More particularly, the disclosed subject matter relates to anew design for a photonic crystal useful for a wide variety ofapplications.

BACKGROUND

Quantum key distribution (QKD) enables two parties to establish a securekey at a distance, even in the presence of one or multipleeavesdroppers. The key can then be used to secretly transmit informationusing unconditionally secure one-time pad encryption. Unconditionalsecurity is achieved by the laws of physics rather than by assumptionsabout the computational abilities of the eavesdropper. Various QKDprotocols have been proposed and implemented. A first protocol dueemployed polarization states of photons passed between the two parties.There has been growing interest in schemes employing photons in Hilbertspaces of high dimension, resulting in a potentially very large alphabetsize. Different degrees of freedom have been considered, includingpolarization, time, and spatial modes.

What is needed is a protocol that allows two parties to generate theirsecure key at the maximum rate allowed using the time-energy basis.Moreover, a protocol is needed that is compatible with fiber-based denseWDM (DWDM) systems commonly used in classical fiber communications. Itis also desirable to have an extremely compact, stable, and scalableplatform for the protocol's implementation.

SUMMARY

A method for distributing a quantum key is provided, including sending afirst photon to a first receiver; sending a second photon to a secondreceiver, the first and second photons being a pair of time-energyentangled photons; and providing a coding scheme comprising a pluralityof time bins and a plurality of frequency bins, wherein a combination ofa time bin and a frequency bin corresponds to a character.

In some embodiments, the first photon and the second photon are sent viaoptical fiber. In some embodiments, the first photon and the secondphoton are sent via a photonic integrated chip. In some embodiments, thefirst photon and the second photon are sent through free space.

A method for distributing a quantum key is provided including generatinga plurality of pairs of time-energy entangled photons; for each pair oftime-energy entangled photons, sending one photon to the first receiverand one photon to the second receiver; and providing a coding schemecomprising a plurality of time bins and a plurality of frequency binswherein each pair of one of the plurality of time bins and one of theplurality of frequency bins corresponds to one of a plurality ofcharacters.

A method of receiving a quantum key is provided including receiving afirst photon, the first photon being one of a pair of time-energyentangled photons; detecting a time of arrival of the first photon;detecting a frequency of the first photon; and determining a characterbased on the detected time and frequency of the first photon.

In some embodiments, the determining a character step further comprises:assigning the detected time to a time bin; assigning the detectedfrequency to a frequency bin; and determining the character from acoding scheme based on the time bin and the frequency bin.

In some embodiments, the method further includes detecting whether aneavesdropper has detected the first photon by determining the frequencydistribution of the first photon.

A method of receiving a quantum key is provided including receiving aplurality of photons, each of the plurality of photons being one of apair of time-energy entangled photons; detecting the time of arrival ofeach of the plurality of photons; detecting the frequency of each of theplurality of photons; and determining a character based on the detectedtime and frequency of each of the plurality of photons.

An apparatus for receiving a quantum key is provided including a channelconfigured to receive photons; a plurality of multi-channel filteringelements, each corresponding to a frequency and each configured to passphotons within a range of its corresponding frequency; and a pluralityof photon detectors, each configured to receive photons passed by one ofthe plurality of multi-channel filtering elements and to send anindication of photon detection.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1( a) illustrates a system using the time-coding scheme inaccordance with exemplary embodiments of the disclosed subject matter.

FIG. 1( b) illustrates time bins in accordance with exemplaryembodiments of the disclosed subject matter.

FIG. 1( c) illustrates a further system using the time-coding scheme inaccordance with exemplary embodiments of the disclosed subject matter.

FIG. 1( d) illustrates time bins in accordance with exemplaryembodiments of the disclosed subject matter.

FIG. 2( a) illustrates Franson interferometers in accordance withexemplary embodiments of the disclosed subject matter.

FIG. 2( b) illustrates the coincidence counting probability P_(C)displayed versus δt in accordance with exemplary embodiments of thedisclosed subject matter.

FIG. 3( a) illustrates mutual information as a function of detectorjitter, σ_(det), normalized to σ_(bin) in accordance with exemplaryembodiments of the disclosed subject matter.

FIG. 3( b) illustrates mutual information as a function of frequencychannels, n_(f) in accordance with exemplary embodiments of thedisclosed subject matter.

FIG. 4 illustrates a further Franson interferometer in accordance withexemplary embodiments of the disclosed subject matter.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS

It is understood that the subject matter described herein is not limitedto particular embodiments described, as such may, of course, vary. It isalso understood that the terminology used herein is for the purpose ofdescribing particular embodiments only, and is not intended to belimiting, since the scope of the present subject matter is limited onlyby the appended claims. Where a range of values is provided, it isunderstood that each intervening value between the upper and lower limitof that range, and any other stated or intervening value in that statedrange, is encompassed within the disclosed subject matter.

Unless defined otherwise, all technical and scientific terms used hereinhave the same meaning as commonly understood by one of ordinary skill inthe art to which this disclosed subject matter belongs. Although anymethods and materials similar or equivalent to those described hereincan also be used in the practice or testing of the present disclosedsubject matter, this disclosure may specifically mention certainexemplary methods and materials.

All publications mentioned in this disclosure are, unless otherwisespecified, incorporated by reference herein for all purposes, including,without limitation, to disclose and describe the methods and/ormaterials in connection with which the publications are cited.

The publications discussed herein are provided solely for theirdisclosure prior to the filing date of the present application. Nothingherein is to be construed as an admission that the present disclosedsubject matter is not entitled to antedate such publication by virtue ofprior invention. Further, the dates of publication provided may bedifferent from the actual publication dates, which may need to beindependently confirmed.

As used herein and in the appended claims, the singular forms “a,” “an”and “the” include plural referents unless the context clearly dictatesotherwise.

Nothing contained in the Abstract or the Summary should be understood aslimiting the scope of the disclosure. The Abstract and the Summary areprovided for bibliographic and convenience purposes and due to theirformats and purposes should not be considered comprehensive.

As will be apparent to those of skill in the art upon reading thisdisclosure, each of the individual embodiments described and illustratedherein has discrete components and features which may be readilyseparated from or combined with the features of any of the other severalembodiments without departing from the scope or spirit of the presentdisclosed subject matter. Any recited method can be carried out in theorder of events recited, or in any other order that is logicallypossible.

Reference to a singular item includes the possibility that there areplural of the same item present. When two or more items (for example,elements or processes) are referenced by an alternative “or,” thisindicates that either could be present separately or any combination ofthem could be present together, except where the presence of onenecessarily excludes the other or others.

As summarized above and as described in further detail below, inaccordance with the various embodiments of the present invention, alarge-alphabet protocol is described herein referred to as a ‘WavelengthDivision Multiplexed Quantum Key Distribution’ (WDM-QKD) that employstime-energy entangled photon pairs. Given a certain photon flux n andchannel bandwidth ΔΩ, WDM-QKD allows two parties, e.g., “Alice and Bob,”to generate their secure key at the maximum rate allowed using thetime-energy basis. WDM-QKD forms the quantum-cryptography analog ofpresent-day WDM systems and is compatible with fiber-based dense WDM(DWDM) systems commonly used in classical fiber communications. Animplementation of WDM-QKD is described herein as both a modern fiberoptic network and in a photonic integrated chip (PIC). The latterprovides an extremely compact, stable, and scalable platform for theprotocol's implementation.

WDM-QKD enables Alice and Bob to generate their shared key at a maximumrate of n log₂(ΔΩ/n) at log₂(ΔΩ/n) bits per photon (bpp), usingpresent-day single photon counters, WDM equipment, and simple componentssuch as beam splitters or directional couplers. The information perphoton per bandwidth is log₂ (ΔΩ/n)/ΔΩ.

QKD protocols derive security from the fact that a measurement changesan unknown state if measured in a conjugate basis. If Alice and Bobemploy two conjugate bases, they can detect a measurement by a thirdparty eavesdropper, “Eve,” including quantum nondemolition measurements.In the protocol described herein, the conjugate bases are time andenergy. Alice and Bob make time and energy measurements using anextended Franson interferometer. Assuming 400 wavelength channels withdetectors having 40 ps jitter, it is expected that for a bandwidth ΔΩ=10nm around 1550 nm, Alice and Bob can generate a key at a maximum of 10Tera bits per second (Tbps) at 1 bit per photon (bpp); or at 10 bpp and100 Gbps; or at 200 Mbps at 20 bpp.

Time-energy entangled photon pairs with correlation time σ_(cor)generated by spontaneous parametric down conversion (SPDC) wereconsidered herein, assuming a pump field at frequency ω_(p) withcoherence time σ_(coh). The photon pair wave function can be written as

|Ψ

=∫_(−∞) ^(∞)∫_(−∞) ^(∞)ψ(t _(A) ,t _(B))|t _(A) ,t _(B) ,o _(A) ,o _(B)

dt _(A) dt _(B)

|t _(A) ,t _(B) ,o _(A) ,o _(B)

=â _(o) _(A) ^(t)(t _(A))â _(o) _(N) ^(t)(t _(N))|0

ψ(t _(A) ,t _(B))∝e ^(−(t) ^(A) ^(−t) ^(B) ⁾ ² ^(/4σ) ^(cor) ² e ^(−t)^(A) ^(+t) ^(B) ⁾ ² ^(/16σ) ^(co) ² e ^(iω) ^(p) ^(/2(t) ^(A) ^(+t) ^(B)⁾  (1)

The creation operator â_(o) _(i) ^(t)(t_(j)) denotes creation in spatialmode o_(i) at time t_(j). If the correlation time of the photons issufficiently smaller than the time bin duration, Alice and Bob canestablish a secret key using the correlated photon arrival time. Aschematic of this protocol is shown in FIG. 1. The continuous biphotonwave function can be discretized as a sum over time bins |σ_(bin) ^(i)

of duration σ_(bin) as

$\begin{matrix}{{{\overset{\_}{\Psi}\rangle} = {\sum\limits_{i = {- \infty}}^{\infty}{\sum\limits_{j = {- \infty}}^{\infty}{G^{i,j}{{\sigma_{bin}^{i},\sigma_{bin}^{j}}\rangle}}}}}{G^{i,j,} = {\int_{i\; \sigma_{bin}}^{{({i + 1})}\sigma_{bin}}{\int_{j\; \sigma_{bin}}^{{({j + 1})}\sigma_{bin}}{{\psi \left( {t_{A},t_{B}} \right)}\ {t_{A}}\ {{t_{B}}.}}}}}} & (2)\end{matrix}$

The probability that Alice and Bob project into time bins σ_(bin) ^(i)and σ_(bin) ^(j) is therefore p^(i,j)=

σ_(bin) ^(i),σ_(bin) ^(j)| ψ

|²=|G^(i,j)|². Detector jitter of magnitude σ_(det) influences thefidelity of this projection, so that roughly σ_(bin)>σ_(det) forreliable communication. Jitter therefore reduces the maximum number ofcharacters to σ_(coh)/σ_(det) from the maximum allowed, given by theSchmidt number, K≈σ_(coh)/σ_(cor). The fastest single photon detectorsprovide σ_(det)≈40 ps, whereas σ_(cor) can be on the order of 10 fs-1ps.

FIG. 1( a) illustrates a system 100 using the time-coding scheme. Astrong laser (not shown) pumps a nonlinear crystal. Photons pairs aregenerated by SPDC 102 and sent across channels of equal length todetectors 106 of Alice's computer 108 and Bob's computer 110 who measuretheir arrival times. FIG. 1( b) illustrates time bins agreed upon byAlice and Bob over a public channel. If a photon pair is detected in agiven time bin, then that character is shared between Alice and Bob.FIG. 1( c) illustrates a system 200 in which photons pairs are alsogenerated by SPDC 202. Alice's computer 208 and Bob's computer 210include a grating, dispersive element or multi-channel filter 212 beforetheir detectors 206 to obtain frequency information. FIG. 1( d)illustrates time bins.

For technological reasons, detector jitter is unlikely to approach thesub-ps regime in the near future. To overcome the technological mismatchbetween photon correlation time and detector jitter, the protocoldescribed herein utilizes the circumstance that photons are not onlyentangled in time, but also in frequency. The biphoton wave function inthe frequency domain, |Ψ_(F)

=FT₂|Ψ

, where FT₂ denotes the two-dimensional Fourier transform. Therefore,|Ψ_(F)

=∫∫

(ω_(A),ω_(B))|ω_(A),ω_(B),o_(A),o_(B)

dω_(A)dω_(B), where

(ω_(A),ω_(B))α exp[−σ_(cor) ²/4(ω_(A)−ω_(B))²]exp[−σ_(coh)²/4(ω_(A)+ω_(B)−ω_(p))²].

States of this form show non-local frequency correlations. Thus, ifAlice measures a frequency ω_(A) on one photon, then Bob must measure afrequency ω_(B)≈ω_(p)−ω_(A) on the other, where ω_(B)=ω_(p)−ω_(A) forσ_(coh)→∞. If both Alice and Bob place multi-channel filtering elementsbefore their detectors (as shown in FIG. 1( c)), then they can collectfrequency information in addition to timing information. This filteringcan be modeled as Gaussian projections onto discrete output spatialmodes |ζ^(i)

with frequency bandwidth δv and center frequency v_(i). Such filteringincreases the correlation time of the photon pair, but this can be keptlower than the detector jitter to minimize the loss of timinginformation. Alternatively, one can increase the time bin durationσ_(bin) to allow for smaller δv. Optimization of these parameters givenlimitations of the source and transmission channel are described below.

The QKD protocol is secure as discussed herein. Measurements of thefrequency and creation time of the biphoton packet disturb its wavefunction in these bases; temporal positive operator valued measurements(POVM) reduce the coherence time while frequency POVMs increase thecorrelation time as discussed in greater detail below.

Alice and Bob can check the security of their channel by testing forchanges in the correlation and coherence times. They use an extendedFranson interferometer (eFI) 300/301, depicted in FIG. 2( a), in whichphotons pairs are also generated by SPDC 302. eFI 300/301 include, e.g.,beam splitters BS 326, partial beam splitters PBS 328, and detectors306. Measurement in Alice's and Bob's arm of the eFI are described bythe annihilation operators for the long and short paths, with respectivetimes t_(L) ^(i) and t_(S) ^(i), and iεA, B denoting Alice and Bob:âA(tA)=1/√{square root over (2)}[âA(t_(L) ^(A))+âA(t_(S) ^(A))] andâB(tB)=1/√{square root over (2)} [âB(t_(L) ^(B))+âA(t_(S) ^(B))].

FIG. 2( a) illustrates for Bob's eFI 301, the short arm is switchedbetween position A and B to achieve lengths δt₁ and δt₂. This allowsdetermination of both and σ_(cor) so that weak frequency and timemeasurements on the photon pair can be detected. FIG. 2( b) illustratesthe coincidence counting probability P_(C) displayed versus δt. Forexample, at position A, δt=0 and P_(c)=1.0.

These times are redefined using standard notation, where Δt=(t_(L)^(A)−t_(S) ^(A)) and δt=(t_(L) ^(A)−t_(S) ^(A))−(t_(L) ^(B)−t_(S) ^(B)).A selection is made to use Δt, Δt−δt>>σ_(cor) in order to avoid singlephoton interference between long and short paths of a single arm of theeFI. Selection of two different values, δt_(1,2), is allowed by placinga switch in the long path of one arm of the eFI. The parameter δt_(1,2)is then scanned to provide phase-dependent coincidence countingmeasurements. Alice and Bob measure the coincidence counting probabilityP_(C), which evaluates to P_(c)∝½+½ cos [ω(2Δt−δt)]e^(−δt) ² ^(/8σ)^(coh) ² e^(−Δt) ² ^(/8σ) ^(coh) ² with visibility V=e^(−δt) ² ^(/8σ)^(cor) ² e^(−Δt) ² ^(/8σ) ^(coh) ² . The correlation time and coherencetime can be deduced from two visibility measurements V₁ and V₂ using δt₁and δt₂, respectively. These extrapolated values σ_(coh) ^(E′) andσ_(cor) ^(E′), are given by

$\begin{matrix}{\mspace{79mu} {\left( \sigma_{cor}^{E^{\prime}} \right)^{2} = {\frac{1}{8}\frac{{\delta \; t_{1}^{2}} - {\delta \; t_{2}^{2}}}{{\ln \; V_{1}} - {\ln \; V_{2}}}}}} & (3) \\{\mspace{79mu} {{\left( \text{?} \right)^{2} = {\frac{1}{8}\frac{\Delta \; t^{2}\left( {{\delta \; t_{1}^{2}} + {\delta \; t_{2}^{2}}} \right)}{{\delta \; t_{1}^{2}\ln \; V_{2}} - {\delta \; t_{2}^{2}\ln \; V_{1}}}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (4)\end{matrix}$

The parameters

δt₁

and

δt₂

are set so that

δt₁

−

δt₂

≈σ_(cor) and |

δt₁

≠|

δt₂

| to sample the Franson curve at different points.

FIGS. 2( a) and 2(b) illustrate the choice of

δt₁

=0 and

δt₂

≈σ_(cor). Using (σ_(coh) ^(E))²=1/[σ_(coh) ^(E′))⁻²−σ_(coh) ⁻²] and(σ_(cor) ^(E))²=1/[(σ_(cor) ^(E′))⁻²−σ_(cor) ⁻²] derived from thismeasurement, the bound on Eve's information per photon isI_(E)≦log₂(σ_(coh)/σ_(coh) ^(E))+log₂(σ_(cor) ^(E)/σ_(cor)), which isthe sum of her information obtained from temporal and frequencymeasurements. While all change in the wave function can be due to Eve'saction, the analysis does not assume that Alice and Bob have perfectdetectors or receive unaltered photon pairs. A more pessimisticcalculation is required for the mutual information between Alice andBob.

The probability of Alice and Bob measuring in frequency channels i and jand time bins k and l, is p^(ijkl)=

ζ_(A) ^(i),ζ_(B) ^(j),σ_(bin,A) ^(k),σ_(bin,B) ¹|Ψ

|²=|G^(ijkl)|². These coefficients form a joint probability densityfunction which can be used to compute the Shannon entropyS=Σ_(ijkl)p^(ijkl) log p^(ijkl) and mutual information I (A,B)=S(A)+S(B)−S(A, B).

$\begin{matrix}{\mspace{79mu} {{{I\left( {A,B} \right)} = {{{- 2}{\sum\limits_{i}^{\;}{{\Gamma^{ik}}^{2}\log {\Gamma^{ik}}^{2}}}} + {\sum\limits_{i,j}^{\;}{{\text{?}}^{2}\log {\text{?}}^{2}}}}}{\Gamma_{ik} = {\int_{k\; \sigma_{bin}}^{{({k + 1})}\sigma_{bin}}{\int_{- \infty}^{\infty}{{{FT}_{2}\left\lbrack {\text{?}{\int_{- \infty}^{\infty}{{\psi \left( {\omega_{A},\omega_{B}} \right)}\ {\omega_{A}}\ {\omega_{B}}}}}\  \right\rbrack}{t_{A}}\ {t_{B}}}}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (5)\end{matrix}$

Detector jitter is included by operating a Gaussian spreading functionon the biphoton state as {circumflex over (σ)}_(det)∫e^(−t) ^(x) ²^(/2σ) ^(det) ² |t

t+t_(x)|dt_(x). FIG. 3 shows the mutual information as a function of thedetector jitter normalized to the time bin duration, as discussedhereinbelow. The joint probability density function P of the new state,|Ψ

={circumflex over (σ)}_(det,A){circumflex over(σ)}_(det,B){Ê}Ŝ_(B)Ŝ_(A)|Ψ_(F)

, incorporating the set of Eve's POVMs, {Ê}, has elements p^(ijkl)=

ζ_(A) ^(k)ζ_(B) ^(l),σ_(bin,A) ^(i),σ_(bin,B) ^(j)|Ψ′

².

FIG. 3( a) illustrates mutual information as a function of detectorjitter, σ_(det), normalized to σ_(bin). FIG. 3( b) illustrates mutualinformation as a function of frequency channels, n_(f). The individualfrequency channels have bandwidth δv=Δω/n_(f) and spacing 4 δv.Increasing the number of frequency channels increases the bits perphoton, until the filter photon temporal bandwidth approaches σ_(bin).

The number of time- and frequency-encoded bits given a photon budget andchannel bandwidth can be optimized. For example, the photon budget isspecific to the photon pair source and is given by the maximum emissionrate, R_(v). The allotted channel frequency bandwidth is Δω. Thefrequency bandwidth of an individual photon pair source is Δω∝1/σ_(cor),so the number of sources to fill the channel bandwidth is N=ΔΩ/Δω. Thecommunication rate, R_(C), in bits per second is therefore

$\begin{matrix}{R_{C} = {{NR}_{v}{\log_{2}\left( \frac{{{\Delta\omega}/\delta}\; v}{R_{v}\sigma_{bin}} \right)}}} & (6)\end{matrix}$

using σ_(bin) and δv defined above.

The often limited photon budget for quantum communication makeshigh-dimensional encoding desirable. However, achieving the limit onthis dimensionality in the time domain using energy-time entangledphoton pairs requires detectors with sub-ps timing jitter andresolution. By invoking conjugate frequency correlations, a protocol toapproach this fundamental limit has been developed using currentdetectors and existing telecom networks. The conjugate nature of timeand energy encoding means that one can trade frequency for temporal bits(and vice versa) to minimize the effect of channel distortion such asnonlinear frequency conversion and dispersion, in addition to optimizingover transmission rate and channel bandwidth.

Gaussian filtering functions are assumed with center frequencies v_(i)and bandwidth δv, which project onto spatial modes |ζ_(i)

. This is summarized in the following relation, where the totalfiltering operator Ŝ is taken as a sum over the individual channelfilters Ŝ^(i).

$\begin{matrix}\begin{matrix}{\hat{S} = {\sum\limits_{i}^{\;}{\hat{S}}_{i}}} \\{= {\sum\limits_{i}^{\;}{\int_{- \infty}^{\infty}{{\exp \left\lbrack \frac{- \left( {\omega - v_{i}} \right)^{2}}{2\delta \; v^{2}} \right\rbrack}{{\omega,\zeta^{i}}\rangle}{\langle{\omega,o}}\ {\omega}}}}}\end{matrix} & (7)\end{matrix}$

The Franson interference derived hereinabove assumes losslesspropagation through the interferometer. However, this assumption is notvalid in photonic integrated chips or fiber networks. Loss can be takeninto account by placing a beam splitter in the long path of the Franson,which couples the waveguide mode with a vacuum mode (see FIG. 4).Working in the Heisenberg construction, evolving the annihilationoperator through the virtual loss beam splitter and the two Franson beamsplitters. The matrix for beam splitters 1 and 2, which leave the thirdmode undisturbed is given by

$\begin{matrix}{{\hat{U}}_{i} = \begin{pmatrix}\sqrt{r_{i}} & \sqrt{1 - r_{i}} & 0 \\\sqrt{1 - r_{i}} & {- \sqrt{r_{i}}} & 0 \\0 & 0 & 1\end{pmatrix}} & (8)\end{matrix}$

where iε1,2. The virtual loss beam splitter is given by

$\begin{matrix}{{\hat{U}}_{L} = \begin{pmatrix}1 & 0 & 0 \\0 & \sqrt{r_{L}} & \sqrt{1 - r_{L}} \\0 & \sqrt{1 - r_{L}} & {- \sqrt{r_{L}}}\end{pmatrix}} & (9)\end{matrix}$

The resulting annihilation operators are thenâ_(A)(t_(A))=C₁â(t)+C₂â(t−Δt) and â_(B)(t_(B))=C₁â(t)+C₂â(t−Δt−δt),disregarding the vacuum term, which will not affect coincidencecounting. C₁=√{square root over (r₁)}√{square root over (r₂)} andC₂=√{square root over (1−r₁)}√{square root over (1−r₂)}√{square rootover (r_(L))}. For maximum visibility, C₁=C₂, so

$\begin{matrix}{\frac{\sqrt{r_{1}}\sqrt{r_{2}}}{\sqrt{1 - r_{1}}\sqrt{1 - r_{2\;}}} = \sqrt{r_{L}}} & (10)\end{matrix}$

This is plotted. For, r₁=r₂=½, the visibility simplifies to

$\begin{matrix}{\mspace{79mu} {{V_{PIC} = {\frac{2\text{?}}{1 + \text{?}}^{{- \delta}\; {t^{2}/2}\sigma_{cor}^{2}}\text{?}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (11)\end{matrix}$

where τ_(a) is the photon lifetime in the waveguide due to loss.

Detector jitter refers to the added uncertainty in the photon detectiontime of some stimulus, purely a result of detector electronics.Superconducting nanowire single photon detectors and InGaAs APDs bothexhibit jitter of roughly 30 to 40 ps. Detector jitter is modeled as aGaussian projection, {circumflex over (σ)}_(det)=∫e^(−t) ^(x) ² ^(/2σ)^(det) ² |t

t+t_(x)|dt_(z). The jitter profile is not truly Gaussian and can bequite asymmetric, however this model allows for first-order analysis. Ifthis is operated on both Alice and Bob's photons, assuming Eq. (1),

$\begin{matrix}{{{\text{?}{\Psi\rangle}} \propto {\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{{\exp \left\lbrack \frac{- \left( {t_{A} + t_{B}} \right)^{2}}{{4\sigma_{\det}^{2}} + {16\sigma_{cab}^{2}}} \right\rbrack}\ {\exp \left\lbrack \frac{- \left( {t_{A} - t_{B}} \right)^{2}}{{4\sigma_{\det}^{2}} + {4\sigma_{cor}^{2}}} \right\rbrack}{{t_{A},t_{B},o_{A},o_{B}}\rangle}{t_{A}}\ {t_{B}}}}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (12)\end{matrix}$

Since σ_(coh)

σ_(det), the most important effect of jitter is to increase the observedcorrelation time roughly from σ_(cor) to σ_(det). This can have asignificant effect on the mutual information between Alice and Bob if∝_(det) is on the order of σ_(bin), as shown in FIG. 3.

The case of a single eavesdropper measuring a single photon of thephoton pair is considered herein. Eve's temporal measurement is a POVMthat can be written as a Gaussian filtering function:

Ê _(t)=∫_(−∞) ^(∞) e ^(−t) ₂ ^(/2(σ) ^(coh) ^(E′) ⁾ ² |

t|dt  (13)

Following, the amplitude function ψ(t_(A),t_(B))αexp[−(t_(A)−t_(B))²/4σ_(cor) ²]exp[−t_(A) ²/4σ_(coh) ²] forσ_(coh)>>σ_(cor). Therefore

$\begin{matrix}{{{\Psi_{E}\rangle} = {{\text{?}{\Psi\rangle}} \propto {\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{{\exp \left\lbrack {- {t_{A}^{2}\left( {\frac{1}{4\sigma_{cab}^{2}} + \frac{1}{4\left( \text{?} \right)^{2}}} \right)}} \right\rbrack}{\exp \left\lbrack \frac{- \left( {t_{A} - t_{B}} \right)^{2}}{\text{?}} \right\rbrack}{{t_{A},t_{B},o_{A},o_{B}}\rangle}{t_{A}}\ {t_{B}}}}}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (14)\end{matrix}$

so the coherence time of the biphoton packet is strongly influenced byEve's filtering bandwidth for σ_(coh) ^(E)>>σ_(coh), which gives thebound on her timing information.

Similarly, a weak frequency POVM is defined,

$\begin{matrix}{\mspace{79mu} {{{\hat{E}}_{\omega} = {\int_{- \infty}^{\infty}{\text{?}{\omega\rangle}{\langle\omega }\ {\omega}}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (15)\end{matrix}$

For 1/σ_(cor)>>1/σ_(coh), and ignoring the spatial modes, |Ψ_(F)

can be written as

|Ψ_(F)

∫∫exp[−σ_(cor) ²/4(2ω_(A)−ω_(p))²]exp[−σ_(coh)²(ω_(A)+ω_(B)−ω_(p))²]|ω_(A),ω_(B)

dω _(A) dω _(B).

Ê _(ω)|Ψ_(F)

≈∫∫exp[−(σ_(cor) ²/4+(σ_(cor) ^(E′))²/4)(2ω_(A)−ω_(p))²]×exp[−σ_(coh)²(ω_(A)+ω_(B)−ω_(p))²]|ω_(A),ω_(B)

dω _(A) dω _(B).  (17)

Thus, Eve projects the biphoton pair onto a narrower frequencydistribution.

As will be apparent to those of skill in the art upon reading thisdisclosure, each of the individual embodiments described and illustratedherein has discrete components and features which may be readilyseparated from or combined with the features of any of the other severalembodiments without departing from the scope or spirit of the presentdisclosed subject matter.

1.-9. (canceled)
 10. An apparatus for receiving a quantum key,comprising: a channel configured to receive photons; a plurality ofmulti-channel filtering elements, each corresponding to a frequency andeach configured to pass photons within a range of its correspondingfrequency; and a plurality of photon detectors, each configured toreceive photons passed by one of the plurality of multi-channelfiltering elements and to send an indication of arrival time of photondetection for each frequency range.
 11. The apparatus of claim 10,wherein the multi-channel filtering element comprises a grating.
 12. Theapparatus of claim 10, wherein the multi-channel filtering elementcomprises a dispersive element.
 13. The apparatus of claim 10, furthercomprising a laser and a non linear crystal.
 14. The apparatus of claim13, wherein the non linear crystal generates pairs of photons byspontaneous parametric down conversion.
 15. The apparatus of claim 14,wherein the pairs of photons are time-energy entangled photon pairs. 16.The apparatus of claim 10, wherein the photons are sent via a photonicintegrated chip.
 17. The apparatus of claim 10, wherein the photons aresent via optical fiber.
 18. An apparatus for receiving a quantum key,comprising: a channel configured to receive photons; a plurality ofmulti-channel filtering elements, each corresponding to a frequency andeach configured to pass photons within a range of its correspondingfrequency; and a plurality of photon detectors, each configured toreceive photons passed by one of the plurality of multi-channelfiltering elements and to determine a time bin based on the arrival timeof photon detection and frequency range.
 19. The apparatus of claim 18,wherein the multi-channel filtering element comprises a grating.
 20. Theapparatus of claim 18, wherein the multi-channel filtering elementcomprises a dispersive element.
 21. The apparatus of claim 18, furthercomprising a laser and a non linear crystal.
 22. The apparatus of claim21, wherein the non linear crystal generates pairs of photons byspontaneous parametric down conversion.
 23. The apparatus of claim 22,wherein the pairs of photons are time-energy entangled photon pairs. 24.The apparatus of claim 18, wherein the photons are sent via a photonicintegrated chip.
 25. The apparatus of claim 18, wherein the photons aresent via optical fiber.
 26. A communication system for distributing aquantum key, comprising: a laser and a non linear crystal adapted togenerate pairs of photons by spontaneous parametric down conversion. afirst apparatus comprising a first channel configured to receivephotons; a first plurality of multi-channel filtering elements, eachcorresponding to a frequency and each configured to pass photons withina range of its corresponding frequency; and a first plurality of photondetectors, each configured to receive photons passed by one of theplurality of multi-channel filtering elements and to send an indicationof arrival time of photon detection for each frequency range; and asecond apparatus comprising a second channel configured to receivephotons; a second plurality of multi-channel filtering elements, eachcorresponding to a frequency and each configured to pass photons withina range of its corresponding frequency; and a second plurality of photondetectors, each configured to receive photons passed by one of theplurality of multi-channel filtering elements and to send an indicationof arrival time of photon detection for each frequency range; and